Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Skewed projections with an application to line stabbing in R3
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Arrangements of lines in 3-space: a data structure with applications
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
A survey of motion planning and related geometric algorithms
Geometric reasoning
Lower bounds on stabbing lines in 3-space
Computational Geometry: Theory and Applications
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Ray shooting and lines in space
Handbook of discrete and computational geometry
Separating a Polyhedron by One Translation from a Set of Obstacles (Extended Abstract)
WG '88 Proceedings of the 14th International Workshop on Graph-Theoretic Concepts in Computer Science
Approximation algorithms for cutting a convex polyhedron out of a sphere
FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management
Cutting out polygons with lines and rays
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Cutting a convex polyhedron out of a sphere
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
On finding a better position of a convex polygon inside a circle to minimize the cutting cost
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Approximation algorithms for cutting a convex polyhedron out of a sphere
Theoretical Computer Science
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We study algorithmic questions related to cutting polyhedral shapes with a hot wire cutter. Such cutters are popular manufacturing tools for cutting expanded polystyrene (styrofoam) with a thin, moving heated wire. In particular, we study the question of polyhedral-wise continuity: Can a given object be cut out without disconnecting and then reattaching the wire? In an abstract setting this question translates to properties of sets of lines and segments and therefore becomes suitable for computational geometry techniques. On the combinatorial and algorithmic levels the results and methods are related to two problems: (1) given a set F= {f1,.....,fk} of polygons and a polygon f, decide if there is a subset of lines in the set of lines not stabbing F that cover f; (2) construct the connectivity graph for free movements of lines that maintain contact with the polyhedral shape. Problem (1) is solved with the dual projection and arrangements of convex and concave x-monotone curves. Problem (2) can be solved with a combination of the skewed projections [6] and hyperbola arrangements proposed by McKenna and O'Rourke [11]. We provide an O(n5) algorithm for constructing a cutting path, if it exists. The complexity of the algorithm is determined by the O(n4) size of the connectivity graph and the cost of solving (2).